The Chile international finally beat Fraser Forster, who had produced a couple of superb saves earlier in the game, to quieten the frustrations of the home support a minute from the end of another game that saw the Gunners struggle in the final third until Olivier Giroud came off the bench.
Defeat was tough on Southampton, who produced a far more solid defensive display than their defeat by Manchester City last time out, and a point would have been a decent reward after finishing the game with 10 men, the superb Toby Alderweireld limping off after Ronald Koeman had made all his substitutions.
After a cagey opening Arsenal created the first chance on six minutes, but Danny Welbeck failed to hit the target after being set free by a lovely Sanchez through-ball.
Southampton’s first opening should really have seen Graziano Pelle end his goalless streak at five games, but the Italy striker, after doing all the hard work to get on the end of a Steven Davis knock-down, smashed his volley way over with only Emiliano Martinez to beat.
Sanchez and Santi Cazorla continued to probe for Arsenal without much in the way of end-product, until midway through the half when Forster was finally forced into a save, getting down quickly to keep out a low Alex Oxlade-Chamberlain drive.
Welbeck almost gave Arsenal the lead in cheeky fashion with a back-heeled effort after a mistake from sub Maya Yoshida but a deflection took the ball wide of Forster’s goal.
Pelle tested Martinez from 25 yards, but the Arsenal keeper was equal to his rising drive, and the best chance of the half eventually fell to the home side in added time, with a glancing header from Welbeck coming off Alderweireld and forcing Forster into a fine reaction save.
Gunners boss Arsene Wenger changed things around at half-time, moving Sanchez up front and Welbeck out to the left, but it was Southampton who started on the front foot with Laurent Koscielny blocking a goalbound effort from the industrious Pelle.
Home frustrations grew as moves continued to break down around the box, Southampton’s usually stingy defence looking solid, and the visitors remained dangerous on the break with Shane Long making a hash of a chance after a superb flowing move from deep.
Cazorla blazed way over the bar, bringing whistles from the crowd, and the collective sighs grew every time a simple pass ended up in touch or in Southampton’s possession.
Wenger introduced Olivier Giroud with 25 minutes left on the clock, for a disappointed Oxlade-Chamberlain, and the Frenchman almost made a quick impact – he was played in by another delicious Sanchez pass but saw his left-foot volley punched away by Forster.
Seconds later, with Arsenal suddenly finding their stride, Giroud’s deft flick set Welbeck up for another chance but the England ‘keeper stretched full-length and tipped his shot around the post.
Southampton continued to create chances on the counter with sub Sadio Mane wasting a decent shooting chance and then having an effort blocked by Per Mertesacker.
They were forced to play the last seven minutes with 10 men, losing the influential Alderweireld – who excelled at the back and later in a defensive midfield role – to what seemed to be a hamstring injury.
Arsenal pushed for a winner and it so nearly came with four minutes remaining, Giroud pulling away to the far post and heading across Forster, only for the ‘keeper to come up with another superb save.
But the vital goal did arrive just a minute later as Sanchez made the most of some rare space in the box to convert an Aaron Ramsey pull back as a number of home players were berating the officials for not giving a penalty for handball.
The wind was firmly out of Southampton’s sails and Arsenal, who climb into sixth in the table, held on comfortably for only their third home win of the season and closed within two points of the Champions League places.
Paul Walsh’s verdict
“Southampton defended brilliantly and you can see why they’ve got a good defensive record. They really frustrated Arsenal. The first half was a really even contest and it really ebbed and flowed, but it seems they made a conscious decision to sit deep on the edge of the box and protect the central area, not allow Arsenal to play through them and play on the counter-attack. In the end, that man Sanchez slid one home right at the death of the game.”
Arsenal: Martinez (7), Chambers (7), Monreal (5), Koscielny (6), Mertesacker (6); Flamini (5), Ramsey (6), Cazorla (6), Sanchez (8), Oxlade-Chamberlain (6), Welbeck (6).
Subs used: Giroud (7), Podolski (6)
Southampton: Forster (9); Clyne (7), Bertrand (7), Fonte (6), Alderweireld (8); Cork (6), Davis (7), Wanyama (7), Tadic (5), Long (5); Pelle (7).
Subs used: Yoshida (6), Mane (5), Mayuka (5).
Man of the match: Fraser Forster.